ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
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The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.Abstract:
We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .
These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.read more
Citations
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Book ChapterDOI
Approximate range searching using binary space partitions
Mark de Berg,Micha Streppel +1 more
TL;DR: It is shown how any BSP tree for the endpoints of a set of n disjoint segments in the plane can be used to obtain a B SP tree of size O(n.depth) such that e-approximate range searching queries with any constant-complexity convex query range can be answered in O(mine>0{1/e+ke}log n) time.
Posted Content
Pseudofinite groups and VC-dimension
Gabriel Conant,Anand Pillay +1 more
TL;DR: Various aspects of “local fsg” for the right-stratified formula are proved for the left invariant NIP formula, which includes a [Formula: see text]-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on [Formulas]: see text].
Dissertation
On some classical and new hypergraph invariants
TL;DR: In this paper, the authors consider several hypergraph parameters and study whether restrictions to subclasses of hypergraphs allow to obtain desirable combinatorial or algorithmic properties, and they provide an optimal theta-bounding function for the class of subcubic triangle-free graphs.
Journal ArticleDOI
From Proximity to Utility: A Voronoi Partition of Pareto Optima
TL;DR: A high-probability, asymptotically optimal bound on the number of Pareto optima points in a point set uniformly sampled from the d-dimensional hypercube is provided and the classical backward analysis technique is revisited, both simplifying and improving relevant results in order to achieve the high- Probability bounds.
Book ChapterDOI
Geometric Hitting Sets for Disks: Theory and Practice
TL;DR: The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects in \(\mathcal{D}\).
References
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Book ChapterDOI
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Book
Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Journal ArticleDOI
On the density of families of sets
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.
Journal ArticleDOI
Central Limit Theorems for Empirical Measures
TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
Journal ArticleDOI
The power of geometric duality
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.